Correspondence theorem for rings

Statement

Let $R$ be a ring, let $I$ be a two-sided ideal of $R$ and let $\pi \colon R \to R/I$ be the quotient map. Let $\mathcal{A}$ be the set of the left ideals $J$ of $R$ with $I \subseteq J$ . Let $\mathcal{B}$ be the set of the left ideals of $R/I$ . The map $\mathcal{A} \to \mathcal{B}$ , $J \mapsto J/I = \pi(J)$ , is a bijection. Its inverse is $K \mapsto \pi^{-1}(K)$ . This bijection and its inverse preserve inclusion.

Remarks

The same bijection holds for right ideals, for two-sided ideals, and for subrings that contain $I$ . For two-sided ideals $J \supseteq I$ , the third isomorphism theorem gives $(R/I)/(J/I) \cong R/J$ . The ideals of $\mathbb{Z}/12\mathbb{Z}$ correspond to the ideals $d\mathbb{Z}$ with $d \mid 12$ , so there are six of them.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Group concept
Ring definition
Ideal definition
Two-sided ideal definition
Quotient ring definition
Correspondence theorem for rings statement

Where this leads

24 pages build on this one.

Tag	`0D1R4SVMBQ`
URI	`m2://0D1R4SVMBQ`
Aliases	`fourth-isomorphism-theorem-for-rings`, `lattice-theorem-for-rings`
Kind	statement, theorem
Writing	W0 stub
MSC	16D25
Status	This copy of the site has no build data. Run `m2 status 0D1R4SVMBQ`.
Uses	Quotient ring, Ideal

Correspondence theorem for rings

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by